Mancini is under pressure after City dropped 12 points behind Manchester United in the title race, but Toure has not given up hope that they can retain their crown, while they have an FA Cup quarter-final against Barnsley to look forward to.
Toure said in the Manchester Evening News: "It's very important for the players, for us to win, because that will help the manager to be here - we don't want to have a season without winning something.
"We delivered last season and we need to maintain that by winning trophies. For a team like City, it's difficult to go for a season without winning a trophy, so for us the FA Cup is really important and the league as well because we want to (keep) this trophy."
Toure's contract expires at the end of the season and he is resigned to leaving the club, but the 31-year-old is hoping to stay in the Premier League.
He said: "I have always liked the Premier League and I want to stay in the top leagues but time goes and if there is no chance I have to go where they want me. There's no chance of staying here but the Premier is my first option for sure."